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Musimathics: The Mathematical Foundations of Music

Gareth Loy
Copyright Date: 2006
Published by: MIT Press
Pages: 504
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  • Book Info
    Book Description:

    A commonsense, self-contained introduction to the mathematics and physics of music; essential reading for musicians, music engineers, and anyone interested in the intersection of art and science.

    eISBN: 978-0-262-27859-1
    Subjects: Mathematics, Music

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-xii)
  3. Foreword
    (pp. xiii-xiv)
    Max Mathews

    Musimathicsby Gareth Loy is a guided tour-de-force of the mathematics and physics of music. It pulls no punches in presenting the scientific fundamentals needed to really understand music, but at the same time it is so clearly written that readers willing to spend time can learn all they need to know to do basic research in modern technical music. Advanced placement courses in math and science in any good high school are plenty of background—from there on Loy leads readers to wherever they want to go.

    Loy has always been a brilliantly clear writer. InMusimathicshe is...

  4. Preface
    (pp. xv-xvi)
  5. About the Author
    (pp. xvi-xvii)
  6. Acknowledgments
    (pp. xvii-xviii)
    Gareth Loy
  7. 1 Music and Sound
    (pp. 1-10)

    The problem of finding the right place to begin an explanation is rather like finding the right fulcrum point to move a stone with a lever. Putting the fulcrum point too close to the stone provides great leverage but little range of movement (figure 1.1a). Putting it too far from the stone provides great range of movement but no leverage (figure 1.1b). The fulcrum point of an explanation is the knowledge and assumptions the reader must already have in order to make sense of the explanation. The assumptions are like the axioms in geometry: a short list of simple, self-evident...

  8. 2 Representing Music
    (pp. 11-38)

    Our ears are continuously bombarded with a stream of pressure fluctuations from the surrounding air, not unlike the way ocean waves ceaselessly beat upon the shore. Nonetheless, our ears discern discrete events in this continuous flow of sound and assign them meaning, such as footsteps, a baby’s cry, or a musical tone.

    Just as the geometrical point is a mental construct that helps us navigate the underlying continuity of space, so the musical tone is a free creation of the human mind that we apply to the unbroken ocean of sound to help us organize and make sense of what...

  9. 3 Musical Scales, Tuning, and Intonation
    (pp. 39-96)

    Why are musical scales organized the way they are? Why is most Western music based on scales made up of seven tones when there are twelve tones per octave? What does “equal-tempered” mean, and why after all these centuries is it still controversial? What choices have other cultures made about intonation, and why? What can we learn about ourselves, our music, and our culture by taking a careful look at the underlying mathematics? This chapter examines one of the most basic issues of music technology: musical scales, tuning, and intonation.

    Certainly, tones and intervals are the primary materials of music....

  10. 4 Physical Basis of Sound
    (pp. 97-128)

    This book uses the international system of standard units defined by the Système International d’Unités, abbreviated SI. It is also known as the MKS system of measurement, which stands for “meter, kilogram, second.” This system is used almost universally by the scientific community as well as by most countries of the world except the United States. As an American, I may occasionally slip back into old habits and use the so-called English “foot, pound, second” system. But since even the English have abandoned it, I’m trying to do so as well.

    The fundamental SI unit of distance is the meter....

  11. 5 Geometrical Basis of Sound
    (pp. 129-148)

    Suppose a pendulum swings back and forth above a turntable. The turntable has a marker, such as a small cone, placed on its surface (figure 5.1). The cone moves withuniform circular motionbecause a motor drives it in a circle at a constant speed. Now adjust the length of the pendulum so that it makes one full swing in the same time that the turntable makes one complete revolution, and release the pendulum at exactly the same moment the cone moves under it so that the two movements are synchronized. With the two motions so aligned, if we look...

  12. 6 Psychophysical Basis of Sound
    (pp. 149-198)

    Suppose you and I were about to play a duet. In order to start, I might signal you by saying, “Ready? One, two, three….”

    For there to be a signal, there must be a source, a receiver, time, distance, and a medium—in this case, air—which spans the distance and connects the source to the receiver. Altogether, this constitutes asignaling system.A signalis a physically detectable quantity such as the pressure of an acoustical wave that traverses a signaling system. More generally, a signal is a description of how any one parameter varies with any other parameter....

  13. 7 Introduction to Acoustics
    (pp. 199-238)

    Having focused on the listener in the previous chapter, I now focus on the medium and consider how sound travels.

    The sounds we hear correspond to pressure disturbances in the medium we are immersed in—air or water. In chapter 6 I mentioned that sound implies a source, a medium, and a receiver. This raises the age-old question: If a tree falls in the forest and there’s no one to hear it, did it make a sound?

    One could argue that pressure disturbances in air are not sound until a subject experiences them, but this seems academic. A way out...

  14. 8 Vibrating Systems
    (pp. 239-284)

    The basis for the production of music and sound lies in the principles of mechanical physics. The physical laws of vibration are highly applicable to music, because they determine not only the sounds instruments make but also how the basilar membrane vibrates in response (see section 6.2.4).

    In section 1.2.2 I broached the subject of one-dimensional harmonic motion of a spring and weight system. In chapter 5 I related it to circular motion. Now it’s time for a still deeper view. Vibration arises from the interaction of an elastic force and inertia. We saw, for example, that these determine the...

  15. 9 Composition and Methodology
    (pp. 285-408)

    The best view of musical composition is provided by a study of methodology. So understanding methodology is the first aim of this chapter. The subject of methodology encompasses most human activities, including the arts and sciences. Approaching composition this way has the great advantage of enabling us to relate the arts and sciences, to see their similarities and differences in sharp relief.

    Studying the methodology of composition provides a crisp and efficient way to identify and compare the aesthetic aims of particular composers and schools of composition. This is of great benefit because we can then accurately compare and contrast...

  16. A Appendix
    (pp. 409-420)
  17. B Appendix
    (pp. 421-452)
  18. Glossary
    (pp. 453-458)
  19. Notes
    (pp. 459-464)
  20. References
    (pp. 465-472)
  21. Equation Index
    (pp. 473-474)
  22. Subject Index
    (pp. 475-482)